In David J. Griffiths' Introduction to Electrodynamics, he claims that the superposition principle is not obvious but has always been found to be consistent with the experiments. So I was wondering have we found some physics quantities which do not follow superposition principle? If we have not till now why can't we generalize and make it into a law?

More specifically: Griffiths was talking about electromagnetic force. My question is about the existence of something like mass or charge and which doesn't follow this superposition principle.

Could you include a quote - without it, it's hard to guess what Griffiths was talking about - maybe the linearity of Maxwells equations, or the superposition of quantum states...?
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twistor59Dec 23 '12 at 14:27

@sree: Learn physics and you will see which quantities can be naturally added and which cannot.
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Vladimir KalitvianskiDec 23 '12 at 17:04

I was unfortunately taught out of Griffiths too. The course had a lot of quirks, the most egregious being this insistence that theory was unable to logically do anything, and that every single, separate equation in physics needed to be verified experimentally, which is of course ridiculous. Superposition is just a part of the theory, and you can't accept electrodynamics as formulated without accepting it.
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Chris WhiteDec 23 '12 at 18:46

4 Answers
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There are plenty of quantities that do not obey the superposition principle. A simple pendulum, for example, will behave differently (with a longer period) if you double the initial amplitude.

What Griffiths means by that quote is that for the electromagnetic field there are no situations where the fields fail to add linearly. More specifically, the superposition principle is encoded in the linearity of Maxwell's equations, which states that

If $(\mathbf{E}_1(\mathbf{r},t),\mathbf{B}_1(\mathbf{r},t))$ and $(\mathbf{E}_2(\mathbf{r},t),\mathbf{B}_2(\mathbf{r},t))$ are solutions of Maxwell's equations, then $$(\mathbf{E}_1(\mathbf{r},t)+\mathbf{E}_2(\mathbf{r},t),\mathbf{B}_1(\mathbf{r},t)+\mathbf{B}_2(\mathbf{r},t))$$ is also a solution.

This is indeed consistent with experiment, except for two situations:

If the field strength inside a medium exceeds that of its linear response, then the material ("macroscopic") Maxwell equations are no longer a linear problem. This is the bread and butter of nonlinear optics, which describes a broad range of phenomena. However, this is not a failure of Griffith's claim, as the 'microscopic' fields $\mathbf{E}$ and $\mathbf{B}$ are still a linear superpositions of those created by the free and bound charges.

In certain, very careful experiments, it is possible to observe the scattering of light by light. This is explained by Quantum Electrodynamics as the temporary creation and annihilation of virtual particle-antiparticle pairs where the light beams meet, which transfer energy and information from one beam into the other. This does violate the superposition principle as stated above and as meant by Griffiths in his textbook, and it has been observed experimentally. However, outside of very specific experiments specially designed to observe it, this effect is negligible and can be ignored as regards classical electrodynamics. In the quantum version, you have a whole host of such problems to deal with.

The answer is that there are "physics quantities" that do not obey a superposition principle. The energy density of the electric field is proportional to $E^2$ so if $\bf{E}=\bf{E}_1+\bf{E}_2$, the energy density is not the sum of the energy densities due to $E_1$ and $E_2$ separately.

Superposition is far from being a universal principle. Suppose that a physical quantity $f$ depends another physical quantity $x$ and that $f$ obeys a superposition law in $x$ so that $f(x_1+x_2)=f(x_1)+f(x_2)$. Then I can always define $\bar{x}=x^3$. Then I can choose to regard $f$ as linear in $x$ or nonlinear in $\bar{x}$.

Perhaps there is some way to phrase what you are really trying to get at with your question that doesn't fall into these obvious loopholes, but my feeling is that it's wrong to think of superposition as being a "universal law".

Superposition principle is normally valid for weak fields. It is implemented in the Maxwell equations that are linear in fields with constant (field-independent) coefficients.

Deviation from linearity occurs for strong fields (non-lienar equations due to field-dependent coefficients). For example, a dielectric breakdown can be described as an essential change in the dielectric conductivity starting from some field value (threshold behavior).

I think we are speaking of different things. In nonlinear interactions, you are dealing with tensor spaces, which does not change the fact that the abstract field operations do not change.
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Antillar MaximusDec 23 '12 at 15:28

Tensor spaces are not obligatory in our case. Imagine that some fields cannot exist due to breakdown they cause. You cannot simply add field strengths. The discharge is a mechanism of decreasing the field.
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Vladimir KalitvianskiDec 23 '12 at 16:24

Any physical quantity that can be organized as a vector space obeys the superposition principle. I would go as far as to say that the superposition principle arises from the fact that a vector space is closed under the weak operation $+$ of the field $\mathbb{(R,+,\cdot)}$.

Nature as we know it can be organized as abstract mathematical objects, such as groups, rings and fields. If this is true, then the conclusion follows directly from the mathematics. For example, the Poincare group plays a central role in describing the fields you speak of. I will revisit this question, for it is a good one and I need a bit of time to organize my thoughts. :)
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Antillar MaximusDec 23 '12 at 15:23

This is mixing up math and physics. We often model a physical system via a vector space because of some superposition principle. The answer basically says that "superposition holds whenever superposition holds".
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sjasonwDec 23 '12 at 19:18